\(A=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^8}\)
\(\Rightarrow3A=1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^7}\)
\(\Rightarrow3A-A=\left(1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^7}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^2}+...+\dfrac{1}{3^8}\right)\)
\(\Rightarrow2A=1-\dfrac{1}{3^8}=\dfrac{6561-1}{6561}=\dfrac{6560}{6561}\Rightarrow A=\dfrac{3280}{6561}\)
Vậy \(A=\dfrac{3280}{6561}\)
\(A=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^8}\)
\(3A=1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{.1}{3^7}\)
\(\Rightarrow3A-A=\left(1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^7}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^8}\right)\)
\(\Rightarrow2A=1-\dfrac{1}{3^8}=\dfrac{3^8}{3^8}-\dfrac{1}{3^8}=\dfrac{3^8-1}{3^8}=\dfrac{6561-1}{6561}=\dfrac{6560}{6561}\)
\(\Rightarrow A=\dfrac{6560}{6561}:2=\dfrac{6560}{6561}\cdot\dfrac{1}{2}=\dfrac{3280}{6561}\)
Vậy \(A=\dfrac{3280}{6561}\).
\(A=\dfrac{1}{3}+\dfrac{1^2}{3}+\dfrac{1^3}{3}+...+\dfrac{1^8}{3}.\)
\(\Rightarrow3A=1+\dfrac{1}{3}+\dfrac{1^2}{3}+\dfrac{1^3}{3}+...+\dfrac{1^7}{3}\).
\(\Rightarrow3A-A=\left(1+\dfrac{1}{3}+\dfrac{1^2}{3}+\dfrac{1^3}{3}+...+\dfrac{1^7}{3}\right)-\left(\dfrac{1}{3}+\dfrac{1^2}{3}+\dfrac{1^3}{3}+...+\dfrac{1^8}{3}\right).\)
\(\Rightarrow2A=1+\dfrac{1}{3}+\dfrac{1^2}{3}+\dfrac{1^3}{3}+...+\dfrac{1^7}{3}-\dfrac{1}{3}-\dfrac{1^2}{3}-\dfrac{1^3}{3}-...-\dfrac{1^8}{3}.\)
\(\Rightarrow A=1-\dfrac{1^8}{3}.\)
\(\Rightarrow A=\dfrac{6560}{6561}.\)
\(\Rightarrow A=\dfrac{6560}{6561}:2=\dfrac{6560}{6561}.\dfrac{1}{2}=\dfrac{3280}{6561}.\)
